Coordinate projections are continuous

#projection#continuity#coordinates

Statement

Show that the projection $\pi_1:\mathbb{R}^n\to\mathbb{R}$ , $(x_1,\ldots,x_n)\mapsto x_1$ , is continuous.

Hint

$|\pi_1(x)-\pi_1(y)|=|x_1-y_1|\le\|x-y\|_1$ .

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Bound. $|\pi_1(x)-\pi_1(y)|\le\|x-y\|_1$ .

Conclusion. $\pi_1$ is 1-Lipschitz, hence continuous.